Australia is a popular destination. Gone (or nearly) are the associations with convicts from Mother England and bars filled with muscle-bound sheep-shearers (women not allowed). Now if you ask for someone's impression of Australia they're likely to say "cosmopolitan", "outdoor living", "beaches", "scenery", "Great Barrier Reef", "sophisticated" even. However, if you want to go there, unless you are a citizen of New Zealand, you are going to need a visa or travel permit.
There are different visas depending on your reasons for wanting to go to Australia.
For social or recreational reasons including a holiday, sightseeing, visiting family and friends or other short term (less than three months) non business or work related reasons, you can apply for an Electronic Travel Authority (Visitor) or a Tourist visa or a Sponsored Family Visitor visa.

If you wish to participate in Australia's Overseas Student Program, you will need to do a registered course full-time and in order to do that, you will need a Student Visa.
For business visits of less than three months you will need a Business (Short-Stay) visa, which will permit you to visit Australia for the purposes of conferences or business meetings. If you are actually organising a business event or conference, you may need a different visa. For longer term employment in Australia, you will need to be sponsored by a company who are prepared to employ you for up to four years.
If you are between 18 and 30 and fancy working your way around Australia, you can participate in the Working Holiday Program which allows you to take up casual employment.

A visit to Australia for medical consultations or treatment requires that you have Medical Treatment Visa.
You can also get a Retirement Visa if you wish to spend some of your retirement years in Australia and a Transit Visa if you want to stop over on the way to somewhere else.
If you want to immigrate to Australia permanently, you will need to participate in the Migration Program which covers people with special skills who will contribute to the Australian economy and people who have family members already in Australia and who are prepared to sponsor them. There is also a Humanitarian Program for refugees.
If you are permanently resident in Australia but are not an Australian citizen, you will need a Resident Return Visa to allow you back in, should you leave on a temporary basis.
Each visa carries different conditions and may require additional paperwork to accompany the application. The one thing all the visas have in common, however, is that you will need to be specific about the length of your proposed stay. If you apply for a visa for three weeks but once you arrive, decide that you want to extend that to two months, you may have a problem.
It is possible that you will have to undergo a medical examination and/or chest x-ray in order to complete your visa application. This is unlikely, however, for a stay of up to 3 months, unless you already have a serious medical condition.

The best place to start for detailed information is the Australian Government Department of Immigration and Multicultural Affairs website.